probability S1 Find the mean and variance of the number of g

S1 Find the mean and variance of the number of games in the World Series. ecall that it is won by the first team to win four games and assume that the outcomes are determined by flipping a coin 51/ Find the mean and variance of the number of games in the Worl

Solution

Let X be the number of games the World Series lasts:

There are 8 possibilities for a World Series.

Team A wins in 4, 5, 6, or 7.
Team B wins in 4, 5, 6, or 7.

The odds that Team A wins a clean sweep is .54=.0625. There\'s exactly the same odds for Team B, so this doubles the probability.

This means P(X=4)=.125

The odds that Team A wins in 5 is found with the binomial distribution, but removing the combination where there were four wins in the first 4 games: (dbinom54dbinom44)(.5)4(.5)1=.125 The odds for Team B are the same, so this doubles the probability.

This means P(X=5)=.25

The odds that Team A wins in 6 is found with the binomial distribution, but removing all combinations where there are 4 wins in the first 5 games: (dbinom64dbinom54)(.5)4(.5)2 Again the odds for Team B are the same which doubles the odds.

This means P(X=6)=.3125

That leaves P(X=7)=1(.125+.25+.3125)=.3125

So mu(X)=(4)(.125)+(5)(.25)+(6)(.3125)+(7)(.3125)

and sigma2(X) =((45.8125)2)(.125) + ((55.8125)2)(.25) + ((65.8125)2)(.3125) + ((75.8125)2)(.3125) =1.02734